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MEN OF MATHEMATICS definite order, corresponding to 'numberable' properties of the elements, the assignment in the given order corresponding to a preassigned ordering of the 'numberable* properties. Granted that this may be even less comprehensible than Riemann's definition, it is nevertheless a -working basis from which to start, and all that it amounts to in plain mathematics is this: a manifold is a set of ordered 'n-tuples' of numbers (aj1Ja?s ... 5 #„), where the parentheses, (), indicate that the numbers £lsa22, ..., xn are to be written in the order given. Two such tt-tuples, (o!lyr2,... , xn) and (y-&& ... , yn) are equal when, and only when, corresponding numbers in them are respectively equal, namely, when, and only when, a^ = ylt #2 = yzi ..,} *n = !/»• If precisely n numbers occur in each of these ordered n-tuples in the manifold, the manifold is said to be of n dimensions. Thus we are back again talking co-ordinates with Descartes. If each of the numbers in (ajlsa2, ... , xn) is a positive, zero, or negative integer, or if it is an element of any countable set (a set whose elements may be counted off 1,2,3, ...), and if the like holds for every n-tuple in the set, the manifold is said to be discrete. If the numbers xvx«, ... , xn, may take on values continuously (as in the motion of a point along a line), the manifold is continuous. This working definition has ignored - deliberately - the ques- tion of whether the set of ordered n-tuples is 'the manifold' or whether something "represented by' these is 'the manifold'. Thus, when we say (x,y) are the co-ordinates of a point in a plane, we do not ask what 'a point in a plane' is, but proceed to work with these ordered couples of numbers (x%) where x,y run through all real numbers independently. On the other hand it may sometimes be advantageous to fix our attention on what such a symbol as (x,y) represents. Thus if # is the age in seconds of a man and y his height in centimetres, we may be interested in the man (or the class of all men) rather than in his co-ordinates with which alone the mathematics of our enquiry is concerned. In this same order of ideas, geometry is no longer concerned with what 'space' 'is' -whether 'is' means anything or not in relation to 'space1. Space, for a modern mathematician, is merely a 55S